Uncertainty quantification for PDEs on hypergraphs

Duration: 09/2023 – 08/2027
Funding scheme: Academy Research Fellow
Funder: Academy of Finland
Budget: 667,532 € + 286,086 €
My role: Principal investigator
Link: https://research.fi/en/results/funding/78093

Public description

3D printing is becoming ubiquitous in engineering and science. One of the main reasons for such success is its ability to create small structures not producible in any other known way. Typical examples include lightweight but strong materials (resembling, e.g., honeycombs) and artificial tissue. Such materials need specific properties, while production is subject to uncertainties in the printing process. This project grows out of the need for mathematical algorithms to find optimal structures that retain their outstanding properties even in the presence of small errors. Robustness is vital for lightweight materials used to build lighter cars, planes, and rockets that save fuel. Similarly, 3D-printed artificial tissue has to mimic the real human tissue of fire victims to a high degree.

Compared to the most efficient existing approaches, the methodology of this project reduces the cost of an optimization-based product design cycle by orders of magnitude.

The picture shows micro-fiber scaffolds that are one artificial tissue component. It is taken from Mechanical behavior of a soft hydrogel reinforced with three-dimensional printed microfibre scaffolds by Miguel Castilho et al., licensed under CC BY 4.0.

Scientific abstract

Modeling such 3D printed objects is not feasible using traditional approaches: On the one hand, they cannot be interpreted as graphs, for which low-cost simulations could be run many times to find an optimal design. On the other hand, the cost of statistics-based optimization using full three-dimensional models is prohibitively high.

This project tackles this issue and gives rise to a new area of mathematics at the interface of applied mathematics and statistics. It uses networks of surfaces, so-called ‘hypergraphs,’ to describe the geometry and topology of 3D printed objects and poses partial differential equations that describe the physical properties of these hypergraphs. We discretize the partial differential equations with hybrid discontinuous Galerkin methods and use Monte Carlo-based statistics to quantify uncertainty. Moreover, we shape-optimize the hypergraph to achieve specific material properties.

Keywords

complex domains, hybrid discontinuous Galerkin, hypergraphs, multilevel Monte Carlo, quasi-Monte Carlo, partial differential equations, singular limits, uncertainty quantification

Related publications

@article{LuWKR23,

title = {Homogeneous multigrid for HDG applied to the Stokes equation},

author = {Peipei Lu and Wei Wang and Guido Kanschat and Andreas Rupp},

url = {https://academic.oup.com/imajna/advance-article-pdf/doi/10.1093/imanum/drad079/52422702/drad079.pdf},

doi = {10.1093/imanum/drad079},

isbn = {0272-4979},

year  = {2023},

date = {2023-10-23},

urldate = {2023-10-23},

journal = {IMA Journal of Numerical Analysis},

abstract = {We propose a multigrid method to solve the linear system of equations arising from a hybrid discontinuous Galerkin (in particular, a single face hybridizable, a hybrid Raviart–Thomas, or a hybrid Brezzi–Douglas–Marini) discretization of a Stokes problem. Our analysis is centered around the augmented Lagrangian approach and we prove uniform convergence in this setting. Beyond this, we establish relations, which resemble those in Cockburn and Gopalakrishnan (2008, Error analysis of variable degree mixed methods for elliptic problems via hybridization. Math. Comput., 74, 1653–1677) for elliptic problems, between the approximates that are obtained by the single-face hybridizable, hybrid Raviart–Thomas and hybrid Brezzi–Douglas–Marini methods. Numerical experiments underline our analytical findings.},

keywords = {},

pubstate = {forthcoming},

tppubtype = {article}

}